Heat oven to 200° F.
Using an electric mixer, beat the egg whites with the vanilla, espresso powder, salt, and cream of tartar on medium-high speed until soft peaks form. Very gradually (1 tablespoon at a time) beat in the sugar; increase mixer speed to high and beat until the mixture holds glossy peaks.
Drop heaping tablespoonfuls of the mixture onto parchment-lined baking sheets, spacing them 1 inch apart.
Bake until dry, crisp, and firm, about 2 hours. (The meringues are done when they release easily from the parchment.)
Turn off the oven, prop open the door about ½ inch (use the handle of a wooden spoon), and let the meringues sit inside for 1 hour. Slide the parchment (with the meringues on top) onto racks and let cool completely.
Peel the meringues off the parchment, then dip the bottom of each into the chocolate, allowing any excess to drip off. Place on parchment-lined baking sheets and refrigerate just until the chocolate is set, 25 to 30 minutes. Store the cookies in an airtight container at room temperature for up to 3 days.
Makes 24 meringues